Limits of discrete structures
Balázs Szegedy Rényi Institute, Budapest
Supported by the ERC grant: Limits of discrete structures The case k = 3 was solved by Roth in 1953 using Fourier analysis. Szemerédi solved the Erdős-Turán conjecture in 1974. 1976: Furstenberg found an analytic approach using measure preser- ving systems. Furstenberg multiple recurrence: Let T :Ω → Ω be an invertible measure preserving transformation on a probability space (Ω, B, µ) and let S ∈ B be a set of positive measure. Then for every k there is d > 0 and x ∈ Ω such that x, T d x, T 2d x,..., T (k−1)d x ∈ S
In 1998 Gowers found an extension of Fourier analysis (called higher order Fourier analysis) that can be used to give explicit bounds in Szemerédi’s theorem.
A far-reaching story in mathematics
Erdős-Turán conjecture (1936): For every k ∈ N and > 0 there is N such that in every subset S ⊆ {1, 2,..., N} with |S|/N ≥ there is a k-term arithmetic progression a, a + b, a + 2b,..., a + (k − 1)b with b 6= 0 contained in S. Szemerédi solved the Erdős-Turán conjecture in 1974. 1976: Furstenberg found an analytic approach using measure preser- ving systems. Furstenberg multiple recurrence: Let T :Ω → Ω be an invertible measure preserving transformation on a probability space (Ω, B, µ) and let S ∈ B be a set of positive measure. Then for every k there is d > 0 and x ∈ Ω such that x, T d x, T 2d x,..., T (k−1)d x ∈ S
In 1998 Gowers found an extension of Fourier analysis (called higher order Fourier analysis) that can be used to give explicit bounds in Szemerédi’s theorem.
A far-reaching story in mathematics
Erdős-Turán conjecture (1936): For every k ∈ N and > 0 there is N such that in every subset S ⊆ {1, 2,..., N} with |S|/N ≥ there is a k-term arithmetic progression a, a + b, a + 2b,..., a + (k − 1)b with b 6= 0 contained in S.
The case k = 3 was solved by Roth in 1953 using Fourier analysis. 1976: Furstenberg found an analytic approach using measure preser- ving systems. Furstenberg multiple recurrence: Let T :Ω → Ω be an invertible measure preserving transformation on a probability space (Ω, B, µ) and let S ∈ B be a set of positive measure. Then for every k there is d > 0 and x ∈ Ω such that x, T d x, T 2d x,..., T (k−1)d x ∈ S
In 1998 Gowers found an extension of Fourier analysis (called higher order Fourier analysis) that can be used to give explicit bounds in Szemerédi’s theorem.
A far-reaching story in mathematics
Erdős-Turán conjecture (1936): For every k ∈ N and > 0 there is N such that in every subset S ⊆ {1, 2,..., N} with |S|/N ≥ there is a k-term arithmetic progression a, a + b, a + 2b,..., a + (k − 1)b with b 6= 0 contained in S.
The case k = 3 was solved by Roth in 1953 using Fourier analysis. Szemerédi solved the Erdős-Turán conjecture in 1974. Furstenberg multiple recurrence: Let T :Ω → Ω be an invertible measure preserving transformation on a probability space (Ω, B, µ) and let S ∈ B be a set of positive measure. Then for every k there is d > 0 and x ∈ Ω such that x, T d x, T 2d x,..., T (k−1)d x ∈ S
In 1998 Gowers found an extension of Fourier analysis (called higher order Fourier analysis) that can be used to give explicit bounds in Szemerédi’s theorem.
A far-reaching story in mathematics
Erdős-Turán conjecture (1936): For every k ∈ N and > 0 there is N such that in every subset S ⊆ {1, 2,..., N} with |S|/N ≥ there is a k-term arithmetic progression a, a + b, a + 2b,..., a + (k − 1)b with b 6= 0 contained in S.
The case k = 3 was solved by Roth in 1953 using Fourier analysis. Szemerédi solved the Erdős-Turán conjecture in 1974. 1976: Furstenberg found an analytic approach using measure preser- ving systems. In 1998 Gowers found an extension of Fourier analysis (called higher order Fourier analysis) that can be used to give explicit bounds in Szemerédi’s theorem.
A far-reaching story in mathematics
Erdős-Turán conjecture (1936): For every k ∈ N and > 0 there is N such that in every subset S ⊆ {1, 2,..., N} with |S|/N ≥ there is a k-term arithmetic progression a, a + b, a + 2b,..., a + (k − 1)b with b 6= 0 contained in S.
The case k = 3 was solved by Roth in 1953 using Fourier analysis. Szemerédi solved the Erdős-Turán conjecture in 1974. 1976: Furstenberg found an analytic approach using measure preser- ving systems. Furstenberg multiple recurrence: Let T :Ω → Ω be an invertible measure preserving transformation on a probability space (Ω, B, µ) and let S ∈ B be a set of positive measure. Then for every k there is d > 0 and x ∈ Ω such that x, T d x, T 2d x,..., T (k−1)d x ∈ S A far-reaching story in mathematics
Erdős-Turán conjecture (1936): For every k ∈ N and > 0 there is N such that in every subset S ⊆ {1, 2,..., N} with |S|/N ≥ there is a k-term arithmetic progression a, a + b, a + 2b,..., a + (k − 1)b with b 6= 0 contained in S.
The case k = 3 was solved by Roth in 1953 using Fourier analysis. Szemerédi solved the Erdős-Turán conjecture in 1974. 1976: Furstenberg found an analytic approach using measure preser- ving systems. Furstenberg multiple recurrence: Let T :Ω → Ω be an invertible measure preserving transformation on a probability space (Ω, B, µ) and let S ∈ B be a set of positive measure. Then for every k there is d > 0 and x ∈ Ω such that x, T d x, T 2d x,..., T (k−1)d x ∈ S
In 1998 Gowers found an extension of Fourier analysis (called higher order Fourier analysis) that can be used to give explicit bounds in Szemerédi’s theorem. Ergodic theory was successfully applied to other hard problems in combinatorics including polynomial and multi dimensional versions of Szemerédi’s theorem. Host-Kra, Ziegler breakthrough results on characteristic factors in er- godic theory clarified the role of geometric-algebraic structures called nilmanifolds in dynamics and in combinatorics. Ideas from higher order Fourier analysis were used in the famous Green-Tao theorem on primes. Szemerédi’s regularity lemma was crucial in the devolpment of dense graph limit theory (2004-) (Chayes, Borgs, Lovász, Sós, Sz., Veszter- gombi) It is an analytic approach to graph theory which leads to new developments in extremal graph theory, property testing, probability theory, etc... The hypergraph regularity method was developed around 2000 by Rödl, Skokan, Nagle, Schacht and Gowers (analytic approach to hy- pergraph regularity: Elek-Sz. 2007)
A far-reaching story in mathematics
The proof of Szemerédi’s theorem lead to the famous Szemerédi regularity lemma which is a fundamental tool in combinatorics. Host-Kra, Ziegler breakthrough results on characteristic factors in er- godic theory clarified the role of geometric-algebraic structures called nilmanifolds in dynamics and in combinatorics. Ideas from higher order Fourier analysis were used in the famous Green-Tao theorem on primes. Szemerédi’s regularity lemma was crucial in the devolpment of dense graph limit theory (2004-) (Chayes, Borgs, Lovász, Sós, Sz., Veszter- gombi) It is an analytic approach to graph theory which leads to new developments in extremal graph theory, property testing, probability theory, etc... The hypergraph regularity method was developed around 2000 by Rödl, Skokan, Nagle, Schacht and Gowers (analytic approach to hy- pergraph regularity: Elek-Sz. 2007)
A far-reaching story in mathematics
The proof of Szemerédi’s theorem lead to the famous Szemerédi regularity lemma which is a fundamental tool in combinatorics. Ergodic theory was successfully applied to other hard problems in combinatorics including polynomial and multi dimensional versions of Szemerédi’s theorem. Ideas from higher order Fourier analysis were used in the famous Green-Tao theorem on primes. Szemerédi’s regularity lemma was crucial in the devolpment of dense graph limit theory (2004-) (Chayes, Borgs, Lovász, Sós, Sz., Veszter- gombi) It is an analytic approach to graph theory which leads to new developments in extremal graph theory, property testing, probability theory, etc... The hypergraph regularity method was developed around 2000 by Rödl, Skokan, Nagle, Schacht and Gowers (analytic approach to hy- pergraph regularity: Elek-Sz. 2007)
A far-reaching story in mathematics
The proof of Szemerédi’s theorem lead to the famous Szemerédi regularity lemma which is a fundamental tool in combinatorics. Ergodic theory was successfully applied to other hard problems in combinatorics including polynomial and multi dimensional versions of Szemerédi’s theorem. Host-Kra, Ziegler breakthrough results on characteristic factors in er- godic theory clarified the role of geometric-algebraic structures called nilmanifolds in dynamics and in combinatorics. Szemerédi’s regularity lemma was crucial in the devolpment of dense graph limit theory (2004-) (Chayes, Borgs, Lovász, Sós, Sz., Veszter- gombi) It is an analytic approach to graph theory which leads to new developments in extremal graph theory, property testing, probability theory, etc... The hypergraph regularity method was developed around 2000 by Rödl, Skokan, Nagle, Schacht and Gowers (analytic approach to hy- pergraph regularity: Elek-Sz. 2007)
A far-reaching story in mathematics
The proof of Szemerédi’s theorem lead to the famous Szemerédi regularity lemma which is a fundamental tool in combinatorics. Ergodic theory was successfully applied to other hard problems in combinatorics including polynomial and multi dimensional versions of Szemerédi’s theorem. Host-Kra, Ziegler breakthrough results on characteristic factors in er- godic theory clarified the role of geometric-algebraic structures called nilmanifolds in dynamics and in combinatorics. Ideas from higher order Fourier analysis were used in the famous Green-Tao theorem on primes. The hypergraph regularity method was developed around 2000 by Rödl, Skokan, Nagle, Schacht and Gowers (analytic approach to hy- pergraph regularity: Elek-Sz. 2007)
A far-reaching story in mathematics
The proof of Szemerédi’s theorem lead to the famous Szemerédi regularity lemma which is a fundamental tool in combinatorics. Ergodic theory was successfully applied to other hard problems in combinatorics including polynomial and multi dimensional versions of Szemerédi’s theorem. Host-Kra, Ziegler breakthrough results on characteristic factors in er- godic theory clarified the role of geometric-algebraic structures called nilmanifolds in dynamics and in combinatorics. Ideas from higher order Fourier analysis were used in the famous Green-Tao theorem on primes. Szemerédi’s regularity lemma was crucial in the devolpment of dense graph limit theory (2004-) (Chayes, Borgs, Lovász, Sós, Sz., Veszter- gombi) It is an analytic approach to graph theory which leads to new developments in extremal graph theory, property testing, probability theory, etc... A far-reaching story in mathematics
The proof of Szemerédi’s theorem lead to the famous Szemerédi regularity lemma which is a fundamental tool in combinatorics. Ergodic theory was successfully applied to other hard problems in combinatorics including polynomial and multi dimensional versions of Szemerédi’s theorem. Host-Kra, Ziegler breakthrough results on characteristic factors in er- godic theory clarified the role of geometric-algebraic structures called nilmanifolds in dynamics and in combinatorics. Ideas from higher order Fourier analysis were used in the famous Green-Tao theorem on primes. Szemerédi’s regularity lemma was crucial in the devolpment of dense graph limit theory (2004-) (Chayes, Borgs, Lovász, Sós, Sz., Veszter- gombi) It is an analytic approach to graph theory which leads to new developments in extremal graph theory, property testing, probability theory, etc... The hypergraph regularity method was developed around 2000 by Rödl, Skokan, Nagle, Schacht and Gowers (analytic approach to hy- pergraph regularity: Elek-Sz. 2007) A far-reaching story in mathematics
Higher order Fourier analysis led to the algebraic theory of nilspaces (Camarena-Sz.) rooted in works of Host and Kra. These objects generalize nilmanifolds and put them into a categorical setting. Nil- spaces can be used to formalize higher order Fourier analysis as an algebraic theory which studies morphisms between nilspaces. Nilspa- ces are also useful in ergodic theory (Candela-Sz.) and in topological dynamic (Gutman-Manners-Varju). We view structures in an asymptotic way. This means that we focus on emergent properties of structures that appear when we look at them on the large scale. (In some sense this is the physics of mathematics) Great program: find the mathematics of large scale behavior A general correspondence: Study of large scale behavior ⇔ study of struc- tural limits
Graphs of different sizes may have a very similar large scale structure. This similarity can be expressed through an appropriate metric. The metric leads to a "limit theory" by taking the completion of the set of structures in the metric. Limit objects in the completion express the "essence" of the large scale behavior. The limit language provides a clean and convenient way to formulate exact statements about asymptotic properties.
Asymptotic view of structures
The above story is very diverse, it has many different aspects. One very dominant common theme is the following: A general correspondence: Study of large scale behavior ⇔ study of struc- tural limits
Graphs of different sizes may have a very similar large scale structure. This similarity can be expressed through an appropriate metric. The metric leads to a "limit theory" by taking the completion of the set of structures in the metric. Limit objects in the completion express the "essence" of the large scale behavior. The limit language provides a clean and convenient way to formulate exact statements about asymptotic properties.
Asymptotic view of structures
The above story is very diverse, it has many different aspects. One very dominant common theme is the following: We view structures in an asymptotic way. This means that we focus on emergent properties of structures that appear when we look at them on the large scale. (In some sense this is the physics of mathematics) Great program: find the mathematics of large scale behavior The metric leads to a "limit theory" by taking the completion of the set of structures in the metric. Limit objects in the completion express the "essence" of the large scale behavior. The limit language provides a clean and convenient way to formulate exact statements about asymptotic properties.
Asymptotic view of structures
The above story is very diverse, it has many different aspects. One very dominant common theme is the following: We view structures in an asymptotic way. This means that we focus on emergent properties of structures that appear when we look at them on the large scale. (In some sense this is the physics of mathematics) Great program: find the mathematics of large scale behavior A general correspondence: Study of large scale behavior ⇔ study of struc- tural limits
Graphs of different sizes may have a very similar large scale structure. This similarity can be expressed through an appropriate metric. The limit language provides a clean and convenient way to formulate exact statements about asymptotic properties.
Asymptotic view of structures
The above story is very diverse, it has many different aspects. One very dominant common theme is the following: We view structures in an asymptotic way. This means that we focus on emergent properties of structures that appear when we look at them on the large scale. (In some sense this is the physics of mathematics) Great program: find the mathematics of large scale behavior A general correspondence: Study of large scale behavior ⇔ study of struc- tural limits
Graphs of different sizes may have a very similar large scale structure. This similarity can be expressed through an appropriate metric. The metric leads to a "limit theory" by taking the completion of the set of structures in the metric. Limit objects in the completion express the "essence" of the large scale behavior. Asymptotic view of structures
The above story is very diverse, it has many different aspects. One very dominant common theme is the following: We view structures in an asymptotic way. This means that we focus on emergent properties of structures that appear when we look at them on the large scale. (In some sense this is the physics of mathematics) Great program: find the mathematics of large scale behavior A general correspondence: Study of large scale behavior ⇔ study of struc- tural limits
Graphs of different sizes may have a very similar large scale structure. This similarity can be expressed through an appropriate metric. The metric leads to a "limit theory" by taking the completion of the set of structures in the metric. Limit objects in the completion express the "essence" of the large scale behavior. The limit language provides a clean and convenient way to formulate exact statements about asymptotic properties. From discrete to continuous
Furstenberg’s approach to Szemerédi’s theorem is a structural limit theory. It can be used to study the asymptotic properties of subsets in integer intervals {1, 2,..., N} as N goes to infinity. Similarity can be defined through pattern frequency. Limit objects are expressed through dynamical systems (Furstenberg’s correspondence principle). Asymptotic behavior becomes precise behavior in the limit. Algebraic and geometric structures such as compact abelian groups and nilmanifolds appear in this limit. Ergodic theory is one of the first examples for an analytic theory that gives a powerful new point of view on discrete structures by expressing asymptotic properties in the frame of a limit theory. From discrete to continuous
For a bounded measurable function W :[0, 1]2 → R let Z kW k := sup W . S,T ⊆[0,1] S×T
Let X be the space of two variable symmetric functions of the form W : [0, 1]2 → [0, 1] and for two functions U, W let
δ (U, W ) := inf kUψ − W k ψ:[0,1]→[0,1]
where ψ is measure preserving and Uψ(x, y) := U(ψ(x), ψ(y)). Note that δ is a pseudo metric. Let U ∼ W if δ (U, W ) = 0. Let G be a graph on the vertex set V (G) = {0, 1,..., n}. We can represent G by the graphon WG defined by
WG (x, y) = 1 if (dxne, dyne) ∈ E(G) and WG (x, y) = 0 otherwise.
Distance of graphs: d(G, H) := δ (W , W ) G H From discrete to continuous
Theorem (Lovász, Sz. 2006): (X / ∼, δ ) is a compact space. Compactness of (X / ∼, δ ) implies two famous results:
Aldous-Hoover theorem on exchangeability (a result in probability theory) Szemerédi’s regularity lemma even in the stronger form by Alon, Fis- cher, Krivelevich, Szegedy.
The space X / ∼ is the completion of the set of finite graphs in δ . Graph limit theory
A "movement" started in the early 2000’s: Bounded degree case: Benjamini, Schramm Reflection positivity: Freedman, Lovász, Schrijver Dense graph limits: Chayes, Borgs, Lovász, Sós, Sz., Vesztergombi Various extensions: Bollobás, Riordan One of the main challenges in graph limit theory is that, depending on the number of edges, there are different useful limit notions. A graph is dense if the number of edges is "quadratic" in the number of vertices. Bounded degree graphs are at the other end of the spectrum. The number of edges is "linear" in the number of vertices. The hardest case is the sub quadratic and super linear case. Some more limit theories
Local-global limits for bounded degree graphs (Bollobás, Riordan, Ha- tami, Lovász, Sz.). This is a refinement of the Benjamini-Schramm limit. Local global limits together with information theory were succ- essfully applied in the study of random regular graph (eigenvector Gaussianity: Backhausz, Sz.). Limits of hypergraphs, and the ultralimit method (Elek, Sz..) Limits of Permutations (Hoppen, Kohayakawa, Moreira, Ráth, Sam- paio) Limits of subsets in abelian groups (Sz.) Note that the Benjamini-Schramm limit theory of colored bounded degree graphs leads to a generalization of Furstenberg’s framework. But there is a very different analytic way of thinking about additive combinatorics which is more similar to dense graph limits!
Limit theory for additive (arithmetic) combinatorics?
The famous correspondence principle by Furstenberg allows us to view ergodic theory of Z-actions as a limit theory for ordered 0-1 sequences. Two such 0-1 sequences are considered to be similar if the frequencies of small consecutive patterns such as 001, 010, 111, etc... are similar Completion in this metric leads to shift invariant Borel measures on the Cantor set {0, 1}Z. Dynamics comes into the picture! Limit theory for additive (arithmetic) combinatorics?
The famous correspondence principle by Furstenberg allows us to view ergodic theory of Z-actions as a limit theory for ordered 0-1 sequences. Two such 0-1 sequences are considered to be similar if the frequencies of small consecutive patterns such as 001, 010, 111, etc... are similar Completion in this metric leads to shift invariant Borel measures on the Cantor set {0, 1}Z. Dynamics comes into the picture! Note that the Benjamini-Schramm limit theory of colored bounded degree graphs leads to a generalization of Furstenberg’s framework. But there is a very different analytic way of thinking about additive combinatorics which is more similar to dense graph limits! What about kf kUk ? What are higher order waves? Gowers’s work started a line of research to understand higher order waves. There are long and deep works by Green, Tao, Ziegler and an alternative approach by me which is also not simple. In my approach higher order wa- ves are basically morphisms from abelian groups to nilmanifolds and more generally to nilspaces. Note that linear characters are homomorphisms to the circle which is the simplest nilmanifold.
Gowers’s work in additive combinatorics
In 1998 Gowers gave a new proof of Szemerédi’s theorem using the so- called Gowers uniformity norms for functions on abelian groups, denoted by
k.kUk . For his method he coined the name Higher order Fourier analysis. A beautiful observation by Gowers is that if a function f : A → C has the
property that kf kU2 is separated enough form 0 then f correlates with a wave (a linear character) on A. Gowers’s work started a line of research to understand higher order waves. There are long and deep works by Green, Tao, Ziegler and an alternative approach by me which is also not simple. In my approach higher order wa- ves are basically morphisms from abelian groups to nilmanifolds and more generally to nilspaces. Note that linear characters are homomorphisms to the circle which is the simplest nilmanifold.
Gowers’s work in additive combinatorics
In 1998 Gowers gave a new proof of Szemerédi’s theorem using the so- called Gowers uniformity norms for functions on abelian groups, denoted by
k.kUk . For his method he coined the name Higher order Fourier analysis. A beautiful observation by Gowers is that if a function f : A → C has the
property that kf kU2 is separated enough form 0 then f correlates with a wave (a linear character) on A.
What about kf kUk ? What are higher order waves? Gowers’s work in additive combinatorics
In 1998 Gowers gave a new proof of Szemerédi’s theorem using the so- called Gowers uniformity norms for functions on abelian groups, denoted by
k.kUk . For his method he coined the name Higher order Fourier analysis. A beautiful observation by Gowers is that if a function f : A → C has the
property that kf kU2 is separated enough form 0 then f correlates with a wave (a linear character) on A.
What about kf kUk ? What are higher order waves? Gowers’s work started a line of research to understand higher order waves. There are long and deep works by Green, Tao, Ziegler and an alternative approach by me which is also not simple. In my approach higher order wa- ves are basically morphisms from abelian groups to nilmanifolds and more generally to nilspaces. Note that linear characters are homomorphisms to the circle which is the simplest nilmanifold. For example let {Xn}n∈N be a system of jointly distributed random variables which is symmetric un- der all permutations of the indices. The famous DeFinetti theorem says that it is a mixture of i.i.d distributions. The Aldous-Hoover theorem cha-
racterizes joint distributions {X } (2) that are symmetric under (n,m) (n,m)∈N (2) the action of SN on N . In this subject symmetry can also mean partial symmetry, Frantzikinakis’s exchangeability result: {Xn}n∈Z, symmetries: shifts and constant multiples. Exchangeability theory is hard in general: only very few cases are solved but all of them are very useful. Tha Aldous-Hoover theorem is closely related to graph theory and graph limit theory. Is there an exchangeability problem which is related to additive and arithmetic combinatorics? YES! There are quite a few
Exchangeability
Exchangeability is a subject in probability which studies joint distributions of random variables with prescribed symmetries. In this subject symmetry can also mean partial
symmetry, Frantzikinakis’s exchangeability result: {Xn}n∈Z, symmetries: shifts and constant multiples. Exchangeability theory is hard in general: only very few cases are solved but all of them are very useful. Tha Aldous-Hoover theorem is closely related to graph theory and graph limit theory. Is there an exchangeability problem which is related to additive and arithmetic combinatorics? YES! There are quite a few
Exchangeability
Exchangeability is a subject in probability which studies joint distributions
of random variables with prescribed symmetries. For example let {Xn}n∈N be a system of jointly distributed random variables which is symmetric un- der all permutations of the indices. The famous DeFinetti theorem says that it is a mixture of i.i.d distributions. The Aldous-Hoover theorem cha-
racterizes joint distributions {X } (2) that are symmetric under (n,m) (n,m)∈N (2) the action of SN on N . Exchangeability theory is hard in general: only very few cases are solved but all of them are very useful. Tha Aldous-Hoover theorem is closely related to graph theory and graph limit theory. Is there an exchangeability problem which is related to additive and arithmetic combinatorics? YES! There are quite a few
Exchangeability
Exchangeability is a subject in probability which studies joint distributions
of random variables with prescribed symmetries. For example let {Xn}n∈N be a system of jointly distributed random variables which is symmetric un- der all permutations of the indices. The famous DeFinetti theorem says that it is a mixture of i.i.d distributions. The Aldous-Hoover theorem cha-
racterizes joint distributions {X } (2) that are symmetric under (n,m) (n,m)∈N (2) the action of SN on N . In this subject symmetry can also mean partial symmetry, Frantzikinakis’s exchangeability result: {Xn}n∈Z, symmetries: shifts and constant multiples. Is there an exchangeability problem which is related to additive and arithmetic combinatorics? YES! There are quite a few
Exchangeability
Exchangeability is a subject in probability which studies joint distributions
of random variables with prescribed symmetries. For example let {Xn}n∈N be a system of jointly distributed random variables which is symmetric un- der all permutations of the indices. The famous DeFinetti theorem says that it is a mixture of i.i.d distributions. The Aldous-Hoover theorem cha-
racterizes joint distributions {X } (2) that are symmetric under (n,m) (n,m)∈N (2) the action of SN on N . In this subject symmetry can also mean partial symmetry, Frantzikinakis’s exchangeability result: {Xn}n∈Z, symmetries: shifts and constant multiples. Exchangeability theory is hard in general: only very few cases are solved but all of them are very useful. Tha Aldous-Hoover theorem is closely related to graph theory and graph limit theory. YES! There are quite a few
Exchangeability
Exchangeability is a subject in probability which studies joint distributions
of random variables with prescribed symmetries. For example let {Xn}n∈N be a system of jointly distributed random variables which is symmetric un- der all permutations of the indices. The famous DeFinetti theorem says that it is a mixture of i.i.d distributions. The Aldous-Hoover theorem cha-
racterizes joint distributions {X } (2) that are symmetric under (n,m) (n,m)∈N (2) the action of SN on N . In this subject symmetry can also mean partial symmetry, Frantzikinakis’s exchangeability result: {Xn}n∈Z, symmetries: shifts and constant multiples. Exchangeability theory is hard in general: only very few cases are solved but all of them are very useful. Tha Aldous-Hoover theorem is closely related to graph theory and graph limit theory. Is there an exchangeability problem which is related to additive and arithmetic combinatorics? Exchangeability
Exchangeability is a subject in probability which studies joint distributions
of random variables with prescribed symmetries. For example let {Xn}n∈N be a system of jointly distributed random variables which is symmetric un- der all permutations of the indices. The famous DeFinetti theorem says that it is a mixture of i.i.d distributions. The Aldous-Hoover theorem cha-
racterizes joint distributions {X } (2) that are symmetric under (n,m) (n,m)∈N (2) the action of SN on N . In this subject symmetry can also mean partial symmetry, Frantzikinakis’s exchangeability result: {Xn}n∈Z, symmetries: shifts and constant multiples. Exchangeability theory is hard in general: only very few cases are solved but all of them are very useful. Tha Aldous-Hoover theorem is closely related to graph theory and graph limit theory. Is there an exchangeability problem which is related to additive and arithmetic combinatorics? YES! There are quite a few Additive (arithmetic) exchangeability
Arithmetic exchangeability is related to the arithmetic groups!
∞ Problem 1: Describe systems of random variables {Xv }v∈Z that are sym- metric under the induced action of GL(∞, Z).
∞ Problem 2 (affine version): Describe systems of random variables {Xv }v∈Z that are symmetric under the induced action of Z∞ o GL(∞, Z). Problem 3 (affine cubic version): Describe systems of random variables {X } ∞ such that for every k ∈ the marginal distributions on v v∈{0,1}0 N affine subcubes are all the same. ∞ ∞ For v ∈ Z let Xv be the random variable (1S ◦ψ)(v). Since hom(Z , A) ∞ is symmetric under the action of GL(∞, Z) the system {Xv }v∈Z is exc- hangeable. Similarly one can construct exchangeble systems satisfying problem 2) and problem 3). Method: Encode functions on Abelian groups into exchangeable systems. In the space of exchangeable systems use weak limits. The limit object is again an exchangeable system. However for proper limit theory a charac- terization of exchangeable systems is missing.
Some explanation: relationship to additive combinatorics
Let A be a finite Abelian group. Let S ⊂ A be a subset in A. The set hom(Z∞, A) is a compact Abelian group so there is a Haar measure. Consequently we can talk about random homomorphisms from Z∞ to A. Method: Encode functions on Abelian groups into exchangeable systems. In the space of exchangeable systems use weak limits. The limit object is again an exchangeable system. However for proper limit theory a charac- terization of exchangeable systems is missing.
Some explanation: relationship to additive combinatorics
Let A be a finite Abelian group. Let S ⊂ A be a subset in A. The set hom(Z∞, A) is a compact Abelian group so there is a Haar measure. Consequently we can talk about random homomorphisms from Z∞ to A. ∞ ∞ For v ∈ Z let Xv be the random variable (1S ◦ψ)(v). Since hom(Z , A) ∞ is symmetric under the action of GL(∞, Z) the system {Xv }v∈Z is exc- hangeable. Similarly one can construct exchangeble systems satisfying problem 2) and problem 3). Some explanation: relationship to additive combinatorics
Let A be a finite Abelian group. Let S ⊂ A be a subset in A. The set hom(Z∞, A) is a compact Abelian group so there is a Haar measure. Consequently we can talk about random homomorphisms from Z∞ to A. ∞ ∞ For v ∈ Z let Xv be the random variable (1S ◦ψ)(v). Since hom(Z , A) ∞ is symmetric under the action of GL(∞, Z) the system {Xv }v∈Z is exc- hangeable. Similarly one can construct exchangeble systems satisfying problem 2) and problem 3). Method: Encode functions on Abelian groups into exchangeable systems. In the space of exchangeable systems use weak limits. The limit object is again an exchangeable system. However for proper limit theory a charac- terization of exchangeable systems is missing. Good news!
We have a solution for problem 3 and it can be used for many applications. Theorem (Candela-Sz. 2018) Every ergodic affine cubic exchangeable sys- tem comes from a so-called nilspace in a "similar way" as the above Abelian construction. Remarks: 1) Nilspaces are slight generalizations of nilmanifolds. Interesting fact for probabilists: Nilmanifolds come up naturally in probability theory! 2) Ergodic measures form a closed set! (Bauer simplex... recall Bauer- Poulsen duality) 3) The proof is long (paper is 98 pages) but about 30 pages deals with some ground work in probability theory, it studies an abstract theory of couplings which can be of independent interest. 4) Difficulty: extract geometric structures (such as nilmanifolds) from a purely probabilistic situation. For this we use a certain topologization method which is inspired by one of my paper with Lovász "Regularity partitions and the topology of graphons" For every natural number k Host and Kra introduced a uniformity ∞ seminorm k.kU(k) on L (Ω). These norms are closely connected to the Gowers norms.
For each Host-Kra seminorm U(k) there is a σ-algebra Bk such that Bk is Z-invariant (it gives a factor system) and furthermore E(f |Bk ) = 0 if and only if kf kU(k+1) = 0. We have that U(k + 1) is ∞ a norm on L (Ω, Bk , µ).
If k = 1 then B1 is the so-called Kronecker factor known from classical ergodic theory. These factors are known to have the structure of a compact abelian group with a given shift.
The factor Bk turns out to be crucial for many problems in addi- tive combinatorics. In particular it is responsible for the k + 2 term arithmetic progressions.
Some background in ergodic theory: The Host-Kra seminorms and factors
In ergodic theory people study measure preserving systems. An im- portant and classical case is when the infinite cyclic group Z acts on a measure space (Ω, A, µ) in a measure preserving way. For each Host-Kra seminorm U(k) there is a σ-algebra Bk such that Bk is Z-invariant (it gives a factor system) and furthermore E(f |Bk ) = 0 if and only if kf kU(k+1) = 0. We have that U(k + 1) is ∞ a norm on L (Ω, Bk , µ).
If k = 1 then B1 is the so-called Kronecker factor known from classical ergodic theory. These factors are known to have the structure of a compact abelian group with a given shift.
The factor Bk turns out to be crucial for many problems in addi- tive combinatorics. In particular it is responsible for the k + 2 term arithmetic progressions.
Some background in ergodic theory: The Host-Kra seminorms and factors
In ergodic theory people study measure preserving systems. An im- portant and classical case is when the infinite cyclic group Z acts on a measure space (Ω, A, µ) in a measure preserving way. For every natural number k Host and Kra introduced a uniformity ∞ seminorm k.kU(k) on L (Ω). These norms are closely connected to the Gowers norms. If k = 1 then B1 is the so-called Kronecker factor known from classical ergodic theory. These factors are known to have the structure of a compact abelian group with a given shift.
The factor Bk turns out to be crucial for many problems in addi- tive combinatorics. In particular it is responsible for the k + 2 term arithmetic progressions.
Some background in ergodic theory: The Host-Kra seminorms and factors
In ergodic theory people study measure preserving systems. An im- portant and classical case is when the infinite cyclic group Z acts on a measure space (Ω, A, µ) in a measure preserving way. For every natural number k Host and Kra introduced a uniformity ∞ seminorm k.kU(k) on L (Ω). These norms are closely connected to the Gowers norms.
For each Host-Kra seminorm U(k) there is a σ-algebra Bk such that Bk is Z-invariant (it gives a factor system) and furthermore E(f |Bk ) = 0 if and only if kf kU(k+1) = 0. We have that U(k + 1) is ∞ a norm on L (Ω, Bk , µ). The factor Bk turns out to be crucial for many problems in addi- tive combinatorics. In particular it is responsible for the k + 2 term arithmetic progressions.
Some background in ergodic theory: The Host-Kra seminorms and factors
In ergodic theory people study measure preserving systems. An im- portant and classical case is when the infinite cyclic group Z acts on a measure space (Ω, A, µ) in a measure preserving way. For every natural number k Host and Kra introduced a uniformity ∞ seminorm k.kU(k) on L (Ω). These norms are closely connected to the Gowers norms.
For each Host-Kra seminorm U(k) there is a σ-algebra Bk such that Bk is Z-invariant (it gives a factor system) and furthermore E(f |Bk ) = 0 if and only if kf kU(k+1) = 0. We have that U(k + 1) is ∞ a norm on L (Ω, Bk , µ).
If k = 1 then B1 is the so-called Kronecker factor known from classical ergodic theory. These factors are known to have the structure of a compact abelian group with a given shift. Some background in ergodic theory: The Host-Kra seminorms and factors
In ergodic theory people study measure preserving systems. An im- portant and classical case is when the infinite cyclic group Z acts on a measure space (Ω, A, µ) in a measure preserving way. For every natural number k Host and Kra introduced a uniformity ∞ seminorm k.kU(k) on L (Ω). These norms are closely connected to the Gowers norms.
For each Host-Kra seminorm U(k) there is a σ-algebra Bk such that Bk is Z-invariant (it gives a factor system) and furthermore E(f |Bk ) = 0 if and only if kf kU(k+1) = 0. We have that U(k + 1) is ∞ a norm on L (Ω, Bk , µ).
If k = 1 then B1 is the so-called Kronecker factor known from classical ergodic theory. These factors are known to have the structure of a compact abelian group with a given shift.
The factor Bk turns out to be crucial for many problems in addi- tive combinatorics. In particular it is responsible for the k + 2 term arithmetic progressions. Some background in ergodic theory: The Host-Kra theorem
The famous Host-Kra theorem gives a geometric characterization of the factors Bk .
Theorem (Host-Kra): The measure preserving system on the factor Bk is the inverse limit of nil-systems. Nil-systems: Let L be a nilpotent Lie group and let Γ ≤ L be a co-compact subgroup. Let N be the left-coset space of Γ. Note that N is a finite dimensional compact manifold. We have that L acts on N in a measure preserving way. A nil-system is the action of a cyclic subgroup of L on N (ergodicity is assumed). It was proved in a paper by Bergelson, Tao and Ziegler that Host-Kra seminorms can be generalized to other Abelian actions and furthermore it was mentioned that these norms can even be generalized to nilpotent ∞ actions. They analized the case of Zp actions and they obtained finite characteristic inverse theorems for Gowers norms using their results. Ber- gelson, Tao, Ziegler and also Host and Kra prepared a program to study the above factors for general nilpotent actions. Our next results contribute to this program in ergodic theory. Applications of additive exchangeability in ergodic theory
Theorem (Candela-Sz. 2018) Characteristic factors of the Host-Kra semi- norms of countable nilpotent group actions are nilspace systems. Theorem (Candela-Sz. 2018) Characteristic factors of the Host-Kra se- minorms of finitely generated nilpotent actions are inverse limits of nil- systems.
These theorems generalize the celebrated Host-Kra, Ziegler results from Z to arbitrary nilpotent group actions. Remark: The above theorem gives a new proof for the Host-Kra, Ziegler theorems, and this new proof is based more on probability theory than on ergodic theory. Some technicality: In the paper we also prove a variant of the previous the- orem. We characterize affine cubic exchangeable systems with a certain conditional independence property. We call them cubic couplings. Alt- hough it seems like a technical notion, it turns out to be more useful in applications. Back to additive limit theory
A sequence of functions fi : Ai → R on compact abelian groups is conver- gent if the density of every fixed linear configuration converges. A linear configuration is given by an integer matrix M ∈ Zk×n. Its density in a function f : A → R is given by the formula:
k Z Y f (Mi,1x1 + Mi,2x2 + ··· + Mi,nxk ). x1,x2,...,xn ∈A i=1
For example the 3 term arithmetic progression density is given by
Z f (x1)f (x1 + x2)f (x1 + 2x2) x1,x2∈A Higher complexity linear forms have a rather complicated limit theory. Li- mit theory restricted to configurations given by 0−1 matrices in the general setting was developed by Sz. in 2010. Limit objects are functions on nil- spaces. Our current results give a refinement of this limit theory: We can take limits of functions already defined on nilspaces.
Back to additive limit theory
There is a theory of complexity of linear forms developed by Gowers, Green, Tao, Wolf and others. It turns out that if we restrict the limit theory to a given complexity then reduced limit theories can be obtained that are simpler than the general limit theory. For example the complexity 1 case was completely described by a paper of Sz. Limit objects here are again functions on compact abelian groups. Our current results give a refinement of this limit theory: We can take limits of functions already defined on nilspaces.
Back to additive limit theory
There is a theory of complexity of linear forms developed by Gowers, Green, Tao, Wolf and others. It turns out that if we restrict the limit theory to a given complexity then reduced limit theories can be obtained that are simpler than the general limit theory. For example the complexity 1 case was completely described by a paper of Sz. Limit objects here are again functions on compact abelian groups. Higher complexity linear forms have a rather complicated limit theory. Li- mit theory restricted to configurations given by 0−1 matrices in the general setting was developed by Sz. in 2010. Limit objects are functions on nil- spaces. Back to additive limit theory
There is a theory of complexity of linear forms developed by Gowers, Green, Tao, Wolf and others. It turns out that if we restrict the limit theory to a given complexity then reduced limit theories can be obtained that are simpler than the general limit theory. For example the complexity 1 case was completely described by a paper of Sz. Limit objects here are again functions on compact abelian groups. Higher complexity linear forms have a rather complicated limit theory. Li- mit theory restricted to configurations given by 0−1 matrices in the general setting was developed by Sz. in 2010. Limit objects are functions on nil- spaces. Our current results give a refinement of this limit theory: We can take limits of functions already defined on nilspaces. An interesting open problem related to exchangeability
Let G = SL(3, Z) and Let Ω = {(a, b, c) ∈ Z3 : gcd(a, b, c) = 1}. G acts on Ω. What about (G, Ω)-exchangeability? Can we characterize G invariant pro- bability measures on {0, 1}Ω? Two things may help: 1) G has Kazdhan property-T ⇒ ergodic measures are closed (Glasner-Weiss) 2) New results in arithmetic exchangeability suggest that such a characterization is not completely out of reach. Moreover we get a generalization for the inverse theorem to nilmanifolds: Gowers norms can be defined for functions on nilmanifolds and this allows us to generalize the concept of higher order Fourier analysis to them.
Invese therems for Gowers norms using exchangeability
Candela-Sz (2019?) If we use the exchangeability result (in fact we use the more technical vers- ion: cubic couplings) as a black-box then we can get a great simplification in the proof of the inverse theorem for the Gowers norms. Invese therems for Gowers norms using exchangeability
Candela-Sz (2019?) If we use the exchangeability result (in fact we use the more technical vers- ion: cubic couplings) as a black-box then we can get a great simplification in the proof of the inverse theorem for the Gowers norms. Moreover we get a generalization for the inverse theorem to nilmanifolds: Gowers norms can be defined for functions on nilmanifolds and this allows us to generalize the concept of higher order Fourier analysis to them.